Investigation: The open box problem.
Investigation: The open box problem Problem: An open box is to be made from a piece of card. Identical squares are to be cut off the four corners of the card to make the box. (As shown below) Cut off Fold lines Aim: Determine the size or the square cut which makes the volume of the box as large as possible for any given rectangular sheet of card. Plan: To start of with I will be using the trial and improvement method to experiment with different sizes of a square boxes. By doing this I will find out the size of cut off that will leave me with the largest volume inside the box. To find out the volume I will need to know the size of the cut off side and the base length. x = length off the square cut off L = original length off the square card The formula that I will use to work out the volume is: Volume = (L-2X) ²X. The different sizes of cards that I will be using are 10cm, 11cm, 12cm, 13cm and 14cm. I will determine the size of x that will give the highest volume to 2d.p. After finding the highest value of X I will prove that my answer if right by using differentiation. Finally I will try and find a rule that allows me to find the highest value of X for a piece of square card and check that it works with any size of square card. Trail and improvement Size of card - 10cm by 10cm X must be 0<X<5: This is because if X is 0 there would not be a side to fold and if
Open Box Problem.
Open Box Problem Aim During this project I will be determine the size of the square cut which makes the volume of the box as large as possible for any given rectangular sheet of card. What is an Open Box An open box is to be made from a sheet of card. Identical squares are cut off the four corners of the card, as shown below. The card is then folded along the dotted lines to make the box. [image001.gif] [image002.gif] Names of things needed for investigation I will write-up my investigation in Microsoft Word and all formulae shall be calculated on Microsoft Excel and all table and graph will be produce in spreadsheets again in Microsoft Excel. Structure of investigation 1. Evidence: · Table · Graphs · Formulas 2. Evaluation 1. Evidence To obtain evidence I will be used a series of methods: · Table · Graphs · Formulae Part 1, Square I am going investigate 3 different sizes for the square open box. Once I have obtained all information on the 3 sizes I will look for patterns and try to formulate a rule to work out the largest volume for an open box square. The sizes that I will be using are: 1. 20 x 20 2. 40 x 40 3. 25 x 25 Because it is a square the length = width so, we can write this as L=1W therefore there is a ratio 1:1. I am going to begin by investigating a square with a side length of 20cm. Using
Open Box Problem
Introduction In this project, I am aiming to: . Determine the size of square cut from any given square sheet of card which makes the volume of an open top box as large as possible. 2. Determine the size of square cut from any given rectangular sheet of card which makes the volume of the resulting open top box as large as possible 4 squares were cut from the paper (1 from each corner). It was then folded along the lines (see diagram), to make an open top cuboid. Different size squares being cut from the paper each time resulted in a different volume. I spent time ring to calculate the size of the square using trial and improvement. Firstly, I examined the size of cut that gave the largest volume of open box by using squared paper to test out some different sizes of squares and rectangles. I then used Microsoft Excel spreadsheets to calculate the lengths, depths and widths to give me the volume of the open box. I calculated the size of cut that would give me the greatest volume to 3 decimal places. To create the box, the equal size squares are cut from the four corners of the card, and it is then folded along the dotted lines. I then put the resultant data into tables to try and calculate relationships between things such as length and square cut. I then tried to calculate a formula that would give me the size of square that must be cut to give me the optimum volume of
The Open Box Problem
The Open Box Problem Maths Coursework Jawhari Thomas 11.5 An open box is a box missing 1 of its six surfaces it can be made using either a squared or rectangular sheet of card. Identical squares are cut from the four corners of the card this creates the height of the box it is then folded as shown below. The card is folded along the dotted lines to form the box. Stage 1 Stage 2 Stage 3 The aim of this exercise is to find the formulae that will enable someone determine the size of the squares cut from the corners of the sheet of card to give the greatest the volume of the box. I am going to begin the investigation using squares, as this will most probably be easiest. I won't build the boxes I am going to use simple mathematics to work out the volumes. Firstly I am going to use an example of how I will carry out the experiment using a square 20cm in length. Using this sized length will allow me to only cut off each corner up to 9.9cm as otherwise I will cause me to run out of card. I am going to begin by looking at cutting the squares off as whole numbers. To find the volume of any box we must use the formula: V = L * W * H When: V is Volume L is Length W is Width and H is Height I know that the both square's length and width will be equal to 20 but The Length of the box will be equal to Length - 2 * Cut Size of the Square (e.g. 20 - 2 * Cut Size of the
The open box problem
The open box problem Introduction The aim of my algebraic investigation into the open box problem is to determine the size of square cut which makes the volume of the box as large as possible for any given rectangular sheet of card. The problem itself is simple, an open box is made from a sheet of card, identical squares are then cut off each of the four corners, the sheet is then folded to make box. It is my aim to find out the maximum square cut which gives me the maximum volume box. Strategy . Try to find the size of cut-out that will give me the maximum volume of a piece of card 6cm x 6cm, progressing onto algebra. 2. Look for limitations in the results. 3. Devise a plan for overcoming these limitations. 4. Extend the work to find a general formula that will help me to work with all squares and rectangles. Technique In this situation there is no need for us to go even close to a pair of scissors or piece of card, for this investigation I am to use Excel. Excel is a computer programme in which I can input information; it will then calculate this information and give me results for what different size cut outs for different sized card. The set-up of this is quite simple; Size of card is out into cell c3 in this case Volume is calculated With a real box by Size of cut-out in Width x Depth x Height Cells a6, a7 etc... The same happens
The Open Box Problem.
The Open Box Problem An open box is to be made from a sheet of card. Identical squares are cut off the four corners of the card, as shown below. The card is then folded along the dotted lines to make the box. The main aim is to determine the size of the square cut which makes the volume of the box as large as possible for any given rectangular sheet of card. I am going to begin by investigating a square with a side length of 20 cm. Using this side length, the maximum whole number I can cut off each corner is 9cm, as otherwise I would not have any box left. I am going to begin by looking into whole numbers being cut out of the box corners. The formula that needs to be used to get the volume of a box is: Volume = Length * Width * Height If I am to use a square of side length 20cm, then I can calculate the side lengths minus the cut out squares using the following equation. Volume = Length - (2 * Cut Out) * Width - (2 * Cut Out) * Height Using a square, both the length & the width are equal. I am using a length/width of 20cm. I am going to call the cut out "x." Therefore the equation can be changed to: Volume = 20 - (2x) * 18 * x If I were using a cut out of length 1cm, the equation for this would be as follows: Volume = 20 - (2 * 1) * 20 - *(2 * 1) * 1 So we can work out through this method that the volume of a box with corners of 1cm² cut out would be: (20 - 2)
Maximum box investigation
maximum box project Ellie Roy Mrs. Satguru MP2b Maths Introduction: This project is about finding the maximum volume of different boxes by investigating what the length of the corner square is. For part 1 we had to find the maximum volume of a box made from a 20 x 20 cm piece of paper. You had to make a box (without a lid) by cutting squares from the corners. The diagram to the left shows where the corner squares are (they are shaded in grey.) It is the size of these corner squares which impact the volume of the finished box. To solve part 1's problem I tried out different corner square lengths and recorder my results in a table. For part 2 we had to do the same thing, however we had to find the volume of a box made from a 24 x 24 cm, 15 x 15 cm, 10 x 10 cm and 36 x 36 cm piece of paper instead of a 20 x 20 cm paper. We also had to try out different corner square lengths and draw tables to show our results for this as well. Part 3 was a bit harder. We had to try and find a connection between the size of the corners cut out and the size of the original piece of paper. But I did manage to find something. Aim: my aim is to find out the largest possible volume of a box made from a 20x20 cm piece of paper. I also want to find the largest possible volume for a box that is made from a 15x15 cm piece of paper, a 24x24 cm piece of paper, a 10x10 piece of paper and a 36x36
The Open Box Problem
Math Coursework The Open Box Problem An open box is to be made from a sheet of card Identical squares are to be cut off the four corners of the card as shown in the following diagram Task: ) For any sized square sheet of card, INVESTIGATE the size of the cut out square, which makes an open box of largest volume 2) For any sized rectangular sheet of card, INVESTIGATE the size of the cut out square, which makes an open box of largest volume In this section of this investigation I am going to investigate using both trial and improvement and spreadsheets which size cutout will give the largest volume of the box. Knowing that this size could go an infinite number of decimal places I have chosen to go to 3 decimal places for the size of the cut out. After investigating the sizes of the cut-outs for 4 different sizes of squares (S2) I will work out the size in a percentage form of the total area of the initial square (S2) using information I have gained from the 4 sizes of squares (S2). After the finding a percentage I will then investigate the sizes for rectangles. I will use 4 different sizes of rectangles for this and will also find the percentage form of the cut out that will get the larges volume for the open box. Finally I will draw a conclusion for both the square card and the rectangular card. The Square I will first use a 10 x 10 square S X B V 0 8 64 0 2
THE OPEN BOX PROBLEM
THE OPEN BOX PROBLEM An open box is to be made from a sheet of card. Identical squares are cut off the four corners of the card, as shown below. The card is then folded along the dotted lines to make the box. The main aim is to determine the size of the square cut which makes the volume of the box as large as possible for any given rectangular sheet of card, but first I am going to experiment with a square to make it easier for me to investigate rectangles. I am going to begin by investigating a square with a side length of 10 cm. Using this side length, the maximum whole number I can cut off each corner is 4.9cm, as otherwise I would not have any box left. I am going to begin by looking into going up in 0.1cm from 0cm being the cut out of the box corners. The formula that needs to be used to get the volume of a box is: Volume = Length * Width * Height If I am to use a square of side length 10cm, then I can calculate the side lengths minus the cut out squares using the following equation. Volume = Length - (2 * Cut Out) * Width - (2 * Cut Out) * Height Using a square, both the length & the width are equal. I am using a length/width of 10cm. I am going to call the cut out "x." Therefore the equation can be changed to: Volume = 10 - (2x) * 10 - (2x) * x If I were using a cut out of length 1cm, the equation for this would be as follows: Volume = 10 - (2 * 1) * 10 -
The Open Box Problem
The Open Box Problem An open box is to be made from a sheet of card. Identical squares are cut off the four corners of the card, as shown below.(fig 1) FIG 1 The card is then folded along the dotted lines to make the box. The main aim is to determine the size of the square cut which makes the volume of the box as large as possible for any given rectangular sheet of card. I am going to begin by investigating a square with a side length of 20 cm. Using this side length, the maximum whole number I can cut off each corner is 9cm, as otherwise I would not have any box left. I am going to begin by looking into whole numbers being cut out of the box corners. The formula that needs to be used to get the volume of a box is: ( * = multiply) Volume = Length * Width * Height If I am to use a square of side length 20cm, then I can calculate the side lengths minus the cut out squares using the following equation. Volume = Length -- (2 * Cut Out) * Width -- (2 * Cut Out) * Height Using a square, both the length & the width are equal. I am using a length/width of 20cm. I am going to call the cut out ""x."" Therefore the equation can be changed to: Volume = 20 -- (2x) * 20 -- (2x) * Height If I were using a cut out of length 1cm, the equation for this would be as follows: Volume = 20 -- (2 * 1) * 20 -- (2 * 1) * 1 So we can work out through this method that the volume